Bar codes of persistent cohomology and Arrhenius law for p-forms

Dorian Le Peutrec (1) , Francis Nier (2) , Claude Viterbo (3, 4)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LAGA - Laboratoire Analyse, Géométrie et Applications
3 DMA - Département de Mathématiques et Applications - ENS Paris
4 PSL - Université Paris Sciences et Lettres

This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenvalues are asymptotically determined by the lengths of the bar code of the function f. In particular, this proves that these quantities are stable in the C^0 topology on the space of functions. Additionally, our analysis provides a general method for computing the subexponential corrections in a large number of cases.

Mots clés: Witten Laplacians; Persistence; Arrhenius Law; Exponentially small eigenvalues

Domaines: Equations aux dérivées partielles [math.AP]; Topologie algébrique [math.AT]; Géométrie différentielle [math.DG]; Physique mathématique [math-ph]; Théorie spectrale [math.SP]; Probabilités [math.PR]

Dorian Le Peutrec, Francis Nier, Claude Viterbo. Bar codes of persistent cohomology and Arrhenius law for p-forms. 2020. ⟨hal-02471644⟩